Subjective Black Hole

My initial reaction was, “Absolutely! This theory of mine
makes no sense.” Consider a black hole. It’s surface area
is proportional to its entropy, so if entropy was subjective, then two
people would claim the same black hole had different sizes.

Let us clearly show that the notion of subjective entropy is
nonsense. Consider two observers, Alice and Bob. Both observe a black
hole and agree it has the same size and entropy. Together they throw a
box of gas towards the black hole. The difference between Alice and Bob
is that Alice knows more about the gas than Bob does. Alice thinks the
box contains a bunch of cold atoms in a perfectly known state. Bob
thinks that the box contains a bunch of hot atoms in an uncertain state.

So Alice and Bob watch the box drift towards the black hole. As the
box falls in it starts radiating its energy away. At a certain point
Bob says, “See there, the box as been sucked into the black
hole!”. The box has gotten close enough to the black hole that it
has done its fancy gravitational
well interacting
thing. The event horizon has expanded, the entropy has been added
to the black hole.

But Alice says, “No. The box hasn’t entered the black
hole yet. See it is still there.” Alice points vaguely at the
dark object. “Where?”, asks Bob. Alice and Bob see a
photon flash. “There!”, exclaims Alice, “That was
light from one of the atoms giving off energy as it falls closer to the
event horizon.” Bob disagrees, “No, that is just Hawking radiation.”

After a long wait, and many more photons, Bob has seen enough Hawking
radiation that the black hole has shrunk back to its original size.
“The box is now gone,” says Alice, “All of its mass
has been converted to radiation so there is nothing left to enter the
black hole.” Before returning home, Alice and Bob stare at the black hole a moment. Both of them agree on its state once again.

Oops, I set out to show that my subjective entropy is crap, but I
ended up showing how two people could observe the same events happening
with a black hole, even if they disagree about the entropy and size of
the black hole. One person sees a large black hole with hawking
radiation. The other person see a small black hole with a pile of junk
near its event horizon, and the junk radiating energy as it falls in.

I think that subjective entropy is a broken theory, but I am still
looking for a paradox.